Category theory can be helpful in understanding Haskell's type system. There exists a "Haskell category", of which the objects are Haskell types, and the morphisms from types a to b are Haskell functions of type a->b.

Definition of a category

Each arrow in Ar has a
domain, dom , and a codomain, cod , each
chosen from Ob. The notation means is an arrow with domain
and codomain . Further, there is a
function called composition, such that is defined only when the codomain of is
the domain of , and in this case,
has the domain of and the codomain of .

In symbols, if and , then .

Also, for each object , there is an arrow
, (often simply denoted as
or , when there is no chance of
confusion).

Axioms

The following axioms must hold for to be a category:

If then (left and right identity)

If and and , then (associativity)

Examples of categories

Set, the category of sets and set functions.

Mon, the category of monoids and monoid morphisms.

Monoids are themselves one-object categories.

Grp, the category of groups and group morphisms.

Rng, the category of rings and ring morphisms.

Grph, the category of graphs and graph morphisms.

Top, the category of topological spaces and continuous maps.

Preord, the category of preorders and order preserving maps.

CPO, the category of complete partial orders and continuous functions.

Further definitions

Categorical programming

Catamorphisms and related concepts, categorical approach to functional programming, categorical programming. Many materials cited here refer to category theory, so as an introduction to this discipline see the #See also section.

Erik Meijer, Maarten Fokkinga, Ross Paterson: Functional Programming with Bananas, Lenses, Envelopes and Barbed Wire. See also related documents (in the CiteSeer page). Understanding the article does not require knowledge of category theory—the paper is self-contained with regard to understanding catamorphisms, anamorphisms and other related concepts.

Books

Bartosz Milewski Category Theory for Programmers. Series of blog posts turned into a book. Covers many abstractions and constructions starting from basics: category, functor up to kan extensions, topos, enriched categories, F-algebras. There are video recordings with those content: part 1, part II and part III.

Michael Barr and Charles Wells: Toposes, Triples and Theories. The online, freely available book is both an introductory and a detailed description of category theory. It also contains a category-theoretical description of the concept of monad (but calling it a triple instead of monad).

R. F. C. Walters: Categories and Computer Science. Category Theory has, in recent years, become increasingly important and popular in computer science, and many universities now introduce Category Theory as part of the curriculum for undergraduate computer science students. Here, the theory is developed in a straightforward way, and is enriched with many examples from computer science.

Arbib&Manes: Arrow, Structures and Functors - The Categorical Imperative. (c)1975 Academic Press, ISBN 0-12-059060-3. Sadly now out of print but very little prerequisite knowledge is needed. It covers monads and the Yoneda lemma.